If by "originally" you mean under the Articles of Confederation. But under the Constitution's Article 1, Section 8, Clause 4: "establish a Uniform Rule of Naturalization" the power to determine citizenship was a granted power of congress
Also, from Madison's Federalist No. 42
The dissimilarity in the rules of naturalization, has long been remarked as a fault in our system, and as laying a foundation for intricate and delicate questions.(...)The new Constitution has accordingly with great propriety made provision against them, and all others proceeding from the defect of the confederation, on this head, by authorising the general government to establish an uniform rule of naturalization throughout the United States.
Yes, the federal government gets to make the rule, but nowhere does it get the authority to grant citizenship.
Citizenship was an issue 'among the several States', so the general government got to make the rules of naturalization in order for all the States to grant citizenship to any inhabitants in a uniform manner.
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BTW- You snipped out the most explanatory part:
Federalist #42
There is a confusion of language here, which is remarkable. Why the terms free inhabitants are used in one part of the article, free citizens in another, and people in another; or what was meant by superadding to "all privileges and immunities of free citizens," "all the privileges of trade and commerce," cannot easily be determined. It seems to be a construction scarcely avoidable, however, that those who come under the denomination of free inhabitants of a State, although not citizens of such State, are entitled, in every other State, to all the privileges of free citizens of the latter; that is, to greater privileges than they may be entitled to in their own State: so that it may be in the power of a particular State, or rather every State is laid under a necessity, not only to confer the rights of citizenship in other States upon any whom it may admit to such rights within itself, but upon any whom it may allow to become inhabitants within its jurisdiction. But were an exposition of the term "inhabitants" to be admitted which would confine the stipulated privileges to citizens alone, the difficulty is diminished only, not removed.